This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.
Smooth functions
All trigonometric functions listed have period 2 π {\displaystyle 2\pi } , unless otherwise stated. For the following trigonometric functions:
Un is the nth up/down number, Bn is the nth Bernoulli number in Jacobi elliptic functions, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}}| Name | Symbol | Formula 1 | Fourier Series |
|---|---|---|---|
| Sine | sin ( x ) {\displaystyle \sin(x)} | ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}} | sin ( x ) {\displaystyle \sin(x)} |
| cas (mathematics) | cas ( x ) {\displaystyle \operatorname {cas} (x)} | sin ( x ) + cos ( x ) {\displaystyle \sin(x)+\cos(x)} | sin ( x ) + cos ( x ) {\displaystyle \sin(x)+\cos(x)} |
| Cosine | cos ( x ) {\displaystyle \cos(x)} | ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}} | cos ( x ) {\displaystyle \cos(x)} |
| cis (mathematics) | e i x , cis ( x ) {\displaystyle e^{ix},\operatorname {cis} (x)} | cos(x) + i sin(x) | cos ( x ) + i sin ( x ) {\displaystyle \cos(x)+i\sin(x)} |
| Tangent | tan ( x ) {\displaystyle \tan(x)} | sin x cos x = ∑ n = 0 ∞ U 2 n + 1 x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle {\frac {\sin x}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}} | 2 ∑ n = 1 ∞ ( − 1 ) n − 1 sin ( 2 n x ) {\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)} 2 |
| Cotangent | cot ( x ) {\displaystyle \cot(x)} | cos x sin x = ∑ n = 0 ∞ ( − 1 ) n 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! {\displaystyle {\frac {\cos x}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}} | i + 2 i ∑ n = 1 ∞ ( cos 2 n x − i sin 2 n x ) {\displaystyle i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)} |
| Secant | sec ( x ) {\displaystyle \sec(x)} | 1 cos x = ∑ n = 0 ∞ U 2 n x 2 n ( 2 n ) ! {\displaystyle {\frac {1}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}} | - |
| Cosecant | csc ( x ) {\displaystyle \csc(x)} | 1 sin x = ∑ n = 0 ∞ ( − 1 ) n + 1 2 ( 2 2 n − 1 − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! {\displaystyle {\frac {1}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}} | - |
| Exsecant | exsec ( x ) {\displaystyle \operatorname {exsec} (x)} | sec ( x ) − 1 {\displaystyle \sec(x)-1} | - |
| Excosecant | excsc ( x ) {\displaystyle \operatorname {excsc} (x)} | csc ( x ) − 1 {\displaystyle \csc(x)-1} | - |
| Versine | versin ( x ) {\displaystyle \operatorname {versin} (x)} | 1 − cos ( x ) {\displaystyle 1-\cos(x)} | 1 − cos ( x ) {\displaystyle 1-\cos(x)} |
| Vercosine | vercosin ( x ) {\displaystyle \operatorname {vercosin} (x)} | 1 + cos ( x ) {\displaystyle 1+\cos(x)} | 1 + cos ( x ) {\displaystyle 1+\cos(x)} |
| Coversine | coversin ( x ) {\displaystyle \operatorname {coversin} (x)} | 1 − sin ( x ) {\displaystyle 1-\sin(x)} | 1 − sin ( x ) {\displaystyle 1-\sin(x)} |
| Covercosine | covercosin ( x ) {\displaystyle \operatorname {covercosin} (x)} | 1 + sin ( x ) {\displaystyle 1+\sin(x)} | 1 + sin ( x ) {\displaystyle 1+\sin(x)} |
| Haversine | haversin ( x ) {\displaystyle \operatorname {haversin} (x)} | 1 − cos ( x ) 2 {\displaystyle {\frac {1-\cos(x)}{2}}} | 1 2 − 1 2 cos ( x ) {\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\cos(x)} |
| Havercosine | havercosin ( x ) {\displaystyle \operatorname {havercosin} (x)} | 1 + cos ( x ) 2 {\displaystyle {\frac {1+\cos(x)}{2}}} | 1 2 + 1 2 cos ( x ) {\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\cos(x)} |
| Hacoversine | hacoversin ( x ) {\displaystyle \operatorname {hacoversin} (x)} | 1 − sin ( x ) 2 {\displaystyle {\frac {1-\sin(x)}{2}}} | 1 2 − 1 2 sin ( x ) {\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\sin(x)} |
| Hacovercosine | hacovercosin ( x ) {\displaystyle \operatorname {hacovercosin} (x)} | 1 + sin ( x ) 2 {\displaystyle {\frac {1+\sin(x)}{2}}} | 1 2 + 1 2 sin ( x ) {\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)} |
| Jacobi elliptic function sn | sn ( x , m ) {\displaystyle \operatorname {sn} (x,m)} | sin am ( x , m ) {\displaystyle \sin \operatorname {am} (x,m)} | 2 π K ( m ) m ∑ n = 0 ∞ q n + 1 / 2 1 − q 2 n + 1 sin ( 2 n + 1 ) π x 2 K ( m ) {\displaystyle {\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1-q^{2n+1}}}~\sin {\frac {(2n+1)\pi x}{2K(m)}}} |
| Jacobi elliptic function cn | cn ( x , m ) {\displaystyle \operatorname {cn} (x,m)} | cos am ( x , m ) {\displaystyle \cos \operatorname {am} (x,m)} | 2 π K ( m ) m ∑ n = 0 ∞ q n + 1 / 2 1 + q 2 n + 1 cos ( 2 n + 1 ) π x 2 K ( m ) {\displaystyle {\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1+q^{2n+1}}}~\cos {\frac {(2n+1)\pi x}{2K(m)}}} |
| Jacobi elliptic function dn | dn ( x , m ) {\displaystyle \operatorname {dn} (x,m)} | 1 − m sn 2 ( x , m ) {\displaystyle {\sqrt {1-m\operatorname {sn} ^{2}(x,m)}}} | π 2 K ( m ) + 2 π K ( m ) ∑ n = 1 ∞ q n 1 + q 2 n cos n π x K ( m ) {\displaystyle {\frac {\pi }{2K(m)}}+{\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}~\cos {\frac {n\pi x}{K(m)}}} |
| Jacobi elliptic function zn | zn ( x , m ) {\displaystyle \operatorname {zn} (x,m)} | ∫ 0 x [ dn ( t , m ) 2 − E ( m ) K ( m ) ] d t {\displaystyle \int _{0}^{x}\left[\operatorname {dn} (t,m)^{2}-{\frac {E(m)}{K(m)}}\right]dt} | 2 π K ( m ) ∑ n = 1 ∞ q n 1 − q 2 n sin n π x K ( m ) {\displaystyle {\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{2n}}}~\sin {\frac {n\pi x}{K(m)}}} |
| Weierstrass elliptic function | ℘ ( x , Λ ) {\displaystyle \wp (x,\Lambda )} | 1 x 2 + ∑ λ ∈ Λ − { 0 } [ 1 ( x − λ ) 2 − 1 λ 2 ] {\displaystyle {\frac {1}{x^{2}}}+\sum _{\lambda \in \Lambda -\{0\}}\left[{\frac {1}{(x-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right]} | {\displaystyle } |
| Clausen function | Cl 2 ( x ) {\displaystyle \operatorname {Cl} _{2}(x)} | − ∫ 0 x ln | 2 sin t 2 | d t {\displaystyle -\int _{0}^{x}\ln \left|2\sin {\frac {t}{2}}\right|dt} | ∑ k = 1 ∞ sin k x k 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}} |
Non-smooth functions
The following functions have period p {\displaystyle p} and take x {\displaystyle x} as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the floor function of n {\displaystyle n} and sgn {\displaystyle \operatorname {sgn} } is the sign function.
K means Elliptic integral K(m)
| Name | Formula | Limit | Fourier Series | Notes |
|---|---|---|---|---|
| Triangle wave | 4 p ( x − p 2 ⌊ 2 x p + 1 2 ⌋ ) ( − 1 ) ⌊ 2 x p + 1 2 ⌋ {\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }} | lim m → 1 − zs ( 4 K x p − K , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {zs} \left({\frac {4Kx}{p}}-K,m\right)} | 8 π 2 ∑ n o d d ∞ ( − 1 ) ( n − 1 ) / 2 n 2 sin ( 2 π n x p ) {\displaystyle {\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)} | non-continuous first derivative |
| Sawtooth wave | 2 ( x p − ⌊ 1 2 + x p ⌋ ) {\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)} | − lim m → 1 − zn ( 2 K x p + K , m ) {\displaystyle -\lim _{m\rightarrow 1^{-}}\operatorname {zn} \left({\frac {2Kx}{p}}+K,m\right)} | 2 π ∑ n = 1 ∞ ( − 1 ) n − 1 n sin ( 2 π n x p ) {\displaystyle {\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)} | non-continuous |
| Square wave | sgn ( sin 2 π x p ) {\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)} | lim m → 1 − sn ( 4 K x p , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {sn} \left({\frac {4Kx}{p}},m\right)} | 4 π ∑ n o d d ∞ 1 n sin ( 2 π n x p ) {\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)} | non-continuous |
| Pulse wave | H ( cos 2 π x p − cos π t p ) {\displaystyle H\left(\cos {\frac {2\pi x}{p}}-\cos {\frac {\pi t}{p}}\right)} where H {\displaystyle H} is the Heaviside step functiont is how long the pulse stays at 1 | t p + ∑ n = 1 ∞ 2 n π sin ( π n t p ) cos ( 2 π n x p ) {\displaystyle {\frac {t}{p}}+\sum _{n=1}^{\infty }{\frac {2}{n\pi }}\sin \left({\frac {\pi nt}{p}}\right)\cos \left({\frac {2\pi nx}{p}}\right)} | non-continuous | |
| Magnitude of sine wave with amplitude, A, and period, p/2 | A | sin π x p | {\displaystyle A\left|\sin {\frac {\pi x}{p}}\right|} | 4 A 2 π + ∑ n = 1 ∞ 4 A π 1 4 n 2 − 1 cos 2 π n x p {\displaystyle {\frac {4A}{2\pi }}+\sum _{n=1}^{\infty }{\frac {4A}{\pi }}{\frac {1}{4n^{2}-1}}\cos {\frac {2\pi nx}{p}}} 3: p. 193 | non-continuous | |
| Cycloid | p − p cos ( f ( − 1 ) ( 2 π x p ) ) 2 π {\displaystyle {\frac {p-p\cos \left(f^{(-1)}\left({\frac {2\pi x}{p}}\right)\right)}{2\pi }}} given f ( x ) = x − sin ( x ) {\displaystyle f(x)=x-\sin(x)} and f ( − 1 ) ( x ) {\displaystyle f^{(-1)}(x)} is its real-valued inverse. | p π ( 3 4 + ∑ n = 1 ∞ J n ( n ) − J n − 1 ( n ) n cos 2 π n x p ) {\displaystyle {\frac {p}{\pi }}{\biggl (}{\frac {3}{4}}+\sum _{n=1}^{\infty }{\frac {\operatorname {J} _{n}(n)-\operatorname {J} _{n-1}(n)}{n}}\cos {\frac {2\pi nx}{p}}{\biggr )}} where J n ( x ) {\displaystyle \operatorname {J} _{n}(x)} is the Bessel Function of the first kind. | non-continuous first derivative | |
| Dirac comb | ∑ n = − ∞ ∞ δ ( x − n p ) {\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-np)} | lim m → 1 − 2 K ( m ) p π dn ( 2 K x p , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}{\frac {2K(m)}{p\pi }}\operatorname {dn} \left({\frac {2Kx}{p}},m\right)} | 1 p ∑ n = − ∞ ∞ e 2 n π i x p {\displaystyle {\frac {1}{p}}\sum _{n=-\infty }^{\infty }e^{\frac {2n\pi ix}{p}}} | non-continuous |
| Dirichlet function | 1 Q ( x ) = { 1 x ∈ Q 0 x ∉ Q {\displaystyle {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}} | lim m , n → ∞ cos 2 m ( n ! x π ) {\displaystyle \lim _{m,n\rightarrow \infty }\cos ^{2m}(n!x\pi )} | - | non-continuous |
Vector-valued functions
- Epitrochoid
- Epicycloid (special case of the epitrochoid)
- Limaçon (special case of the epitrochoid)
- Hypotrochoid
- Hypocycloid (special case of the hypotrochoid)
- Spirograph (special case of the hypotrochoid)
Doubly periodic functions
Notes
References
Formulae are given as Taylor series or derived from other entries. ↩
Orloff, Jeremy. "ES.1803 Fourier Expansion of tan(x)" (PDF). Massachusetts Institute of Technology. Archived from the original (PDF) on 2019-03-31. https://web.archive.org/web/20190331130103/http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf ↩
Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571. 978-3834807571 ↩